Introduction to Computers and
Programming
Prof. I. K. Lundqvist
Lecture 18
May 3 2004
Four Variable K-Maps
Example-2
Using a 4-variable K-Map, simplify the following Truth table
A
B
C
D
Output
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
1
1
0
0
1
0
0
1
0
1
0
1
0
0
1
1
0
0
0
1
1
1
0
1
0
0
0
0
1
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
1
0
0
0
1
1
0
1
0
1
1
1
0
1
1
1
1
1
0
AB
CD
00
01
11
0
1
0
0
0
0
0
1
11
0
0
0
0
10
0
0
1
0
00
01
10
Output = ABCD + ABCD + ABCD
3
Four Variable K-Maps
Example-2
Using a 4-variable K-Map, simplify the following Truth table
A
B
C
D
Output
0
0
0
0
1
0
0
0
1
1
0
0
1
0
0
0
0
1
1
0
0
1
0
0
1
0
1
0
1
1
0
1
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
0
1
0
1
0
1
0
1
1
0
1
1
0
1
1
0
0
0
1
1
0
1
0
1
1
1
0
1
1
1
1
1
1
AB
CD
00
01
11
10
00
01
11
10
4
Product-of-Sums
from a Truth Table
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F
0
0
0
1
1
1
1
1
F
1
1
1
0
0
0
0
0
Find an expression for F
F = ABC + ABC + ABC
F = ABC + ABC + ABC
F = ABC • ABC • ABC
F = ( A + B + C) • ( A + B + C ) • ( A + B + C)
6
Maxterms
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F
0
0
0
1
1
1
1
1
F
1
1
1
0
0
0
0
0
Maxterms
F = ( A + B + C) • ( A + B + C ) • ( A + B + C)
• To find a Product-of-Sums form for a truth table
– Make one maxterm for each row in which the function is zero
– For each maxterm, each variable appears once
• In its complemented form if it is one in the row
• In its regular form if it is zero in the row
7
Today
• Propositional Logic
• From English to propositions
• Quantified statements
• Tomorrow: Methods of proving theorems
8
Propositional Logic
• Logic at the sentential level
– Smallest unit: sentence
– Sentences that can be either true or false
– This kind of sentences are called
Propositions
• If a propositions is true, then its truth
value is “true”, if proposition if false,
then the truth value is “false”
9
Propositional Logic
• The following are propositions:
– Grass is green
–2+4=4
• The following are not propositions:
–
–
–
–
Wake up
Is it raining today?
X>2
X=X
10
Elements of Propositional Logic
Connectives
not
¬
and
∧
or
∨
if_then
(implies)
iff
→
↔
11
Connectives
P
Q
(P ∨ Q)
P
Q
(P ∧ Q)
P
Q
(P → Q)
F
F
T
T
F
T
F
T
F
T
T
T
F
F
T
T
F
T
F
T
F
F
F
T
F
F
T
T
F
T
F
T
T
T
F
T
P
¬P
P
Q
(P ↔ Q)
T
F
F
T
F
F
T
T
F
T
F
T
T
F
F
T
Truth tables
12
Concept Question
Given P Æ Q, Is Q Æ P True?
1. Yes
2. No
3. I don’t know
4. What is P->Q
13
Converse and Contrapositive
• For P Æ Q
QÆP
¬Q Æ ¬P
is called its converse
is called its contrapositive
Example: If it rains, then I get soaked
converse :
If I get soaked, then it rains
contrapositive :
If I don’t get soaked, then it does not rain
14
From English to Proposition
• Premises:
P
Q
– It snows
– If it snows, then the school is closed
The school is closed
• Rules of inference
[P ∧ (P Æ Q)] Æ Q
15
From English to Proposition
• Restate given statements using building
blocks and the connectives
• Propositions
–P
–Q
–R
it is raining
I will go to the beach
I have time
• “I will go to the beach if it is not
raining” restate “If it is not raining, I
will go to the beach” restate ¬PÆQ
16
Exercise
• Restate: ” I will go to the beach if is not
raining and I have time ”
“If it is not raining and I have time, then
I will go to the beach”
(¬P ∧ R) Æ Q
17
Rule of Inference: Modus Ponens
(p ∧ (p → q)) → q is a tautology. It states that if we
know that both an implication p → q is true and that its
hypothesis, p, is true, then the conclusion, q, is true.
Ex: Suppose the implication “If the bus breaks down, then
I will have to walk” and its hypothesis “the bus breaks
down” are true.
Then by modus ponens it follows that “I will have to walk”.
Ex: Assume that the implication (n > 3) → (n2 > 9) is true.
Suppose also that n > 3.
Then by modus ponens, it follows that n2 > 9.
18
Fallacy: Affirming the Conclusion
(q ∧ (p → q)) → p is a contingency. It states that if we
know that both an implication p → q is true and that its
conclusion, q, is true, then the hypothesis, p, is true.
Ex: Suppose the implication “If the bus breaks down,
then I will have to walk” and its conclusion “I will have
to walk” is true. It does not follow that the bus broke
down. Perhaps I simply missed the bus.
Ex: Consider the implication (n > 3) → (n2 > 9) which is
true. Suppose also that n2 > 9. It does not follow that
n > 3. It might be that n = -4 for example.
19